dgegv (3) - Linux Manuals

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NAME

dgegv.f -

SYNOPSIS

Functions/Subroutines

subroutine dgegv (JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO)
DGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Function/Subroutine Documentation

subroutine dgegv (characterJOBVL, characterJOBVR, integerN, double precision, dimension( lda, * )A, integerLDA, double precision, dimension( ldb, * )B, integerLDB, double precision, dimension( * )ALPHAR, double precision, dimension( * )ALPHAI, double precision, dimension( * )BETA, double precision, dimension( ldvl, * )VL, integerLDVL, double precision, dimension( ldvr, * )VR, integerLDVR, double precision, dimension( * )WORK, integerLWORK, integerINFO)

DGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Purpose:

 This routine is deprecated and has been replaced by routine DGGEV.

 DGEGV computes the eigenvalues and, optionally, the left and/or right
 eigenvectors of a real matrix pair (A,B).
 Given two square matrices A and B,
 the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
 eigenvalues lambda and corresponding (non-zero) eigenvectors x such
 that

    A*x = lambda*B*x.

 An alternate form is to find the eigenvalues mu and corresponding
 eigenvectors y such that

    mu*A*y = B*y.

 These two forms are equivalent with mu = 1/lambda and x = y if
 neither lambda nor mu is zero.  In order to deal with the case that
 lambda or mu is zero or small, two values alpha and beta are returned
 for each eigenvalue, such that lambda = alpha/beta and
 mu = beta/alpha.

 The vectors x and y in the above equations are right eigenvectors of
 the matrix pair (A,B).  Vectors u and v satisfying

    u**H*A = lambda*u**H*B  or  mu*v**H*A = v**H*B

 are left eigenvectors of (A,B).

 Note: this routine performs "full balancing" on A and B

Parameters:

JOBVL

          JOBVL is CHARACTER*1
          = 'N':  do not compute the left generalized eigenvectors;
          = 'V':  compute the left generalized eigenvectors (returned
                  in VL).

JOBVR

          JOBVR is CHARACTER*1
          = 'N':  do not compute the right generalized eigenvectors;
          = 'V':  compute the right generalized eigenvectors (returned
                  in VR).

          N is INTEGER
          The order of the matrices A, B, VL, and VR.  N >= 0.

          A is DOUBLE PRECISION array, dimension (LDA, N)
          On entry, the matrix A.
          If JOBVL = 'V' or JOBVR = 'V', then on exit A
          contains the real Schur form of A from the generalized Schur
          factorization of the pair (A,B) after balancing.
          If no eigenvectors were computed, then only the diagonal
          blocks from the Schur form will be correct.  See DGGHRD and
          DHGEQZ for details.

LDA

          LDA is INTEGER
          The leading dimension of A.  LDA >= max(1,N).

          B is DOUBLE PRECISION array, dimension (LDB, N)
          On entry, the matrix B.
          If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
          upper triangular matrix obtained from B in the generalized
          Schur factorization of the pair (A,B) after balancing.
          If no eigenvectors were computed, then only those elements of
          B corresponding to the diagonal blocks from the Schur form of
          A will be correct.  See DGGHRD and DHGEQZ for details.

LDB

          LDB is INTEGER
          The leading dimension of B.  LDB >= max(1,N).

ALPHAR

          ALPHAR is DOUBLE PRECISION array, dimension (N)
          The real parts of each scalar alpha defining an eigenvalue of
          GNEP.

ALPHAI

          ALPHAI is DOUBLE PRECISION array, dimension (N)
          The imaginary parts of each scalar alpha defining an
          eigenvalue of GNEP.  If ALPHAI(j) is zero, then the j-th
          eigenvalue is real; if positive, then the j-th and
          (j+1)-st eigenvalues are a complex conjugate pair, with
          ALPHAI(j+1) = -ALPHAI(j).

BETA

          BETA is DOUBLE PRECISION array, dimension (N)
          The scalars beta that define the eigenvalues of GNEP.
          
          Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
          beta = BETA(j) represent the j-th eigenvalue of the matrix
          pair (A,B), in one of the forms lambda = alpha/beta or
          mu = beta/alpha.  Since either lambda or mu may overflow,
          they should not, in general, be computed.

          VL is DOUBLE PRECISION array, dimension (LDVL,N)
          If JOBVL = 'V', the left eigenvectors u(j) are stored
          in the columns of VL, in the same order as their eigenvalues.
          If the j-th eigenvalue is real, then u(j) = VL(:,j).
          If the j-th and (j+1)-st eigenvalues form a complex conjugate
          pair, then
             u(j) = VL(:,j) + i*VL(:,j+1)
          and
            u(j+1) = VL(:,j) - i*VL(:,j+1).

          Each eigenvector is scaled so that its largest component has
          abs(real part) + abs(imag. part) = 1, except for eigenvectors
          corresponding to an eigenvalue with alpha = beta = 0, which
          are set to zero.
          Not referenced if JOBVL = 'N'.

LDVL

          LDVL is INTEGER
          The leading dimension of the matrix VL. LDVL >= 1, and
          if JOBVL = 'V', LDVL >= N.

          VR is DOUBLE PRECISION array, dimension (LDVR,N)
          If JOBVR = 'V', the right eigenvectors x(j) are stored
          in the columns of VR, in the same order as their eigenvalues.
          If the j-th eigenvalue is real, then x(j) = VR(:,j).
          If the j-th and (j+1)-st eigenvalues form a complex conjugate
          pair, then
            x(j) = VR(:,j) + i*VR(:,j+1)
          and
            x(j+1) = VR(:,j) - i*VR(:,j+1).

          Each eigenvector is scaled so that its largest component has
          abs(real part) + abs(imag. part) = 1, except for eigenvalues
          corresponding to an eigenvalue with alpha = beta = 0, which
          are set to zero.
          Not referenced if JOBVR = 'N'.

LDVR

          LDVR is INTEGER
          The leading dimension of the matrix VR. LDVR >= 1, and
          if JOBVR = 'V', LDVR >= N.

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.  LWORK >= max(1,8*N).
          For good performance, LWORK must generally be larger.
          To compute the optimal value of LWORK, call ILAENV to get
          blocksizes (for DGEQRF, DORMQR, and DORGQR.)  Then compute:
          NB  -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR;
          The optimal LWORK is:
              2*N + MAX( 6*N, N*(NB+1) ).

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1,...,N:
                The QZ iteration failed.  No eigenvectors have been
                calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
                should be correct for j=INFO+1,...,N.
          > N:  errors that usually indicate LAPACK problems:
                =N+1: error return from DGGBAL
                =N+2: error return from DGEQRF
                =N+3: error return from DORMQR
                =N+4: error return from DORGQR
                =N+5: error return from DGGHRD
                =N+6: error return from DHGEQZ (other than failed
                                                iteration)
                =N+7: error return from DTGEVC
                =N+8: error return from DGGBAK (computing VL)
                =N+9: error return from DGGBAK (computing VR)
                =N+10: error return from DLASCL (various calls)

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

: November 2011

Further Details:

  Balancing
  ---------

  This driver calls DGGBAL to both permute and scale rows and columns
  of A and B.  The permutations PL and PR are chosen so that PL*A*PR
  and PL*B*R will be upper triangular except for the diagonal blocks
  A(i:j,i:j) and B(i:j,i:j), with i and j as close together as
  possible.  The diagonal scaling matrices DL and DR are chosen so
  that the pair  DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to
  one (except for the elements that start out zero.)

  After the eigenvalues and eigenvectors of the balanced matrices
  have been computed, DGGBAK transforms the eigenvectors back to what
  they would have been (in perfect arithmetic) if they had not been
  balanced.

  Contents of A and B on Exit
  -------- -- - --- - -- ----

  If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
  both), then on exit the arrays A and B will contain the real Schur
  form[*] of the "balanced" versions of A and B.  If no eigenvectors
  are computed, then only the diagonal blocks will be correct.

  [*] See DHGEQZ, DGEGS, or read the book "Matrix Computations",
      by Golub & van Loan, pub. by Johns Hopkins U. Press.

Definition at line 306 of file dgegv.f.

Author

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